Can he buy meat from a non-Muslim butcher who says that it is halaal?

Firstly: it is better for you to buy the meat you need from
Muslims, because this is more on the safe side and there is less likely to
be any doubt concerning it. You will also be helping the Muslims to continue
their work for which there is a need in the western countries. Moreover, the
Muslims undoubtedly deserve this business more than others. It is sufficient
for the Muslim to feel that he is renewing his bonds of faith with his
Muslim brothers every time he forsakes the stores that are owned by kuffaar
that may be closer to him or sell goods for cheaper prices, to buy from his
brothers in faith. See question no.
9205. 

Secondly: If the non-Muslim butcher is not from among the
People of the Book, then meat slaughtered by him is not halaal. If he is
from among the People of the Book, Jewish or Christian, then meat
slaughtered by him is halaal. 

Ibn Qudaamah said: The scholars are unanimously agreed that
it is permissible to eat meat slaughtered by the People of the Book, because
Allaah says (interpretation of the meaning): “The
food (slaughtered cattle, eatable animals) of the people of the Scripture
(Jews and Christians) is lawful to you” [al-Maa’idah 5:5].
Al-Bukhaari said: Ibn ‘Abbaas said: “Their food” means meat slaughtered by
them. Al-Mughni, 13/293. 

After that we do not need
to ask them about the way in which they slaughtered it, because the basic
principle is that their slaughter is valid because it was done by them as
People of the Book. See question no.
20805. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

It is proven in Saheeh al-Bukhaari (no. 5507) from
‘Aa’ishah (may Allaah be pleased with her) that some people came to the
Prophet (peace and blessings of Allaah be upon him) and said: “Some
people bring meat to us, and we do not know if they mentioned the name of
Allaah or not.” He said: “Say it yourselves then eat.” 

I [Shaykh Ibn ‘Uthaymeen
is speaking] say: They were new in Islam, and they did not know whether they
had said the name of Allaah or not, so he said: “Say it yourselves, then
eat.” So eating it is permissible even if we do not know whether the one who
slaughtered the meat mentioned the name of Allaah or not. Similarly it is
permissible to eat even if we do not know whether the meat was slaughtered
in the proper manner or not, because if the action is carried out by the
right people then the basic principle is that it is valid, unless there is
evidence to the contrary. So if some meat comes to us from a Muslim, a Jew
or a Christian, we should not ask about it or ask how it was slaughtered or
whether the name of Allaah was mentioned over it. It is halaal so long as
there is no clear proof that it is haraam. This is a way in which Allaah has
made things easier for us. Liqaa’aat al-Baab il-Maftooh, 1/77 

It is clear from this that if you buy from this store and eat
the food of the one from whom you buy it, there is nothing wrong with that
and you do not need to ask him about the way in which the meat was
slaughtered, unless you become certain that he has slaughtered it in a
manner that is not prescribed, such as stunning the animal until it dies and
so on. But it is better to buy from the Muslim as stated above. 

And Allaah is the Source of strength.