If an eclipse happens at the time of Jumu‘ah prayer, which prayer should be done first?

Firstly: 

If the sun is eclipsed on a Friday, if that occurs before
Jumu‘ah at a time when there is sufficient time to do the ordinary eclipse
prayer (salaat al-kusoof), such as if the eclipse occurs in the forenoon or
close to it, then you should start with the eclipse prayer, then pray
Jumu‘ah at the proper time. If the eclipse occurs at the time of Jumu‘ah, if
there is the fear that Jumu‘ah prayer may be missed, then Jumu‘ah should be
done first, according to scholarly consensus. 

If there is no risk of it being missed, then the majority of
scholars are of the view that the eclipse prayer should be done first. The
Hanbalis follow the view that was favoured by Ibn Qudaamah (may Allah have
mercy on him), that Jumu‘ah should be done first, because starting with the
eclipse prayer may lead to hardship for some people, because it will lead to
keeping people back for the sake of praying the eclipse prayer and
compelling them to pray it, when it is not obligatory in the first place. 

Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni
(2/146):

if two prayers occur at the same time, such as the eclipse
prayer with another prayer such as Jumu‘ah or Eid or one of the obligatory
prayers or Witr, then people should start with the one which they are more
afraid of missing. If there is the fear that they may miss both, then they
should start with the obligatory prayer. If neither of them is obligatory,
such as the eclipse prayer and Witr or Taraweeh, then they should start with
the one which is more emphasised. For example, in the case of the eclipse
prayer and Witr, they should start with the eclipse prayer, because it is
more emphasised, and for that reason it is Sunnah to offer it in
congregation; moreover, Witr may be made up later on but the eclipse prayer
cannot be made up later on. 

If Taraweeh and the eclipse prayer coincide, then which of
them should be done first? There are two points of view concerning this. The
view of our companions, and the correct view in my opinion, is that the
obligatory prayer that is offered in congregation takes precedence over the
eclipse prayer in all situations, because giving precedence to the eclipse
prayer over it will cause hardship and will require the people present to
offer the eclipse prayer even though it is not obligatory upon them, and
will force them to wait for the obligatory prayer, even though among them
there may be those who are physically weak, elderly or have needs to attend
to. The Prophet (blessings and peace of Allah be upon him) instructed us to
make the obligatory prayer brief, so that it would not cause undue hardship
to those who are praying in congregation behind the imam. Hence it is more
appropriate to say that we should not cause hardship by offering this
lengthy and difficult prayer, even though it is not obligatory. The same
ruling applies if it coincides with Taraweeh prayer; Taraweeh should be
given precedence for the same reason. If the eclipse prayer coincides with
Witr at the beginning of the time for Witr, it should be given precedence,
because Witr  will not be missed; if there is the fear that Witr will be
missed, precedence should be given to Witr, because it is brief and can be
done easily, and one can still catch up with the time of the eclipse prayer.
If there is no time left except as much is needed to do Witr, then there is
no need to undertake the eclipse prayer, because it has occurred at a time
when prayer is not allowed. If the eclipse coincides with the time of a
funeral prayer, the funeral prayer should be given precedence according to
scholarly consensus, because there is concern for the deceased. And Allah
knows best. End quote. 

An-Nawawi (may Allah have mercy on him) said in al-Majmoo‘
(5/61): ash-Shaafa‘i and his companions (may Allah have mercy on them) said:
When two prayers coincide, precedence should be given to the one that it is
feared may be missed, then that which is most emphasised. If Eid and the
eclipse prayer coincide, or Jumu‘ah and the eclipse prayer, and there is the
fear that the Eid prayer or Jumu‘ah prayer may be missed because there is
not enough time, then precedence should be given to the Eid or Jumu‘ah
prayer, because they are more emphasised than the eclipse prayer. If there
is no fear of missing them, then the more correct view, which was stated
categorically by the author [Abu Ishaaq ash-Shiraazi] and the majority, is
that precedence should be given to the eclipse prayer, because there is the
fear that it may be missed. End quote. 

See also: al-Mawsoo‘ah al-Foqhiyyah (27/258) 

What appears to be more correct is the view of Ibn Qudaamah
(may Allah have mercy on him), because of what he mentioned about difficulty
and hardship, and because Jumu‘ah is more emphasised and more important. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was
asked: If two prayers coincide, the eclipse prayer and another prayer, such
as an obligatory prayer, or Jumu‘ah prayer, or Witr, or Taraweeh, which
should be given precedence? 

He replied: 

The obligatory prayer should be given precedence over the
eclipse prayer (for solar or lunar eclipse), because it is more important,
and because Allah, may He be exalted, said in the hadith qudsi: “My slave
does not draw close to Me by means of anything more beloved to Me than that
which I have made obligatory for him.”

End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen (16/307) 

Secondly: 

If the imam begins with Jumu‘ah, he should deliver the
khutbah then lead the Jumu‘ah prayer, then he should offer the eclipse
prayer and deliver a khutbah for that. If he starts with the eclipse prayer
and completes it, he should deliver a khutbah for Jumu‘ah and mention the
eclipse in it, then pray Jumu‘ah, and the two Jumu‘ah khutbahs will suffice
for the khutbah that is usually given with the eclipse prayer. 

An-Nawawi (may Allah have mercy on him) said, in the book
quoted from above: If Jumu‘ah and an eclipse coincide, and the situation
requires giving precedence to Jumu‘ah, he should deliver the khutbah for it
then pray Jumu‘ah, then offer the eclipse prayer, then give a khutbah for
the eclipse. 

If the situation requires giving precedence to the eclipse
prayer, he should start with it, then give the khutbah for Jumu‘ah, making
mention of the eclipse in it and what is recommended in this situation in
the two Jumu‘ah khutbahs, and there is no need for four khutbahs. Our
companions said: This refers to the two Jumu‘ah khutbahs only. This was also
stated by ash-Shaafa‘i in al-Umm. End quote. 

And Allah knows best.