I was on a J1 visa for a year, it has now expired I want to return on an VWP

One point in your favour is that you left the US (22nd of September 2016) before your J-1 visa expired (30th of September 2016).

"possibly travel around" however does not sound convincing. It is not only vague but having been in USA, even as a conscientious student, for more than a year (since August 2015) it begs questions such as how it was you did not make the opportunity to travel around while already in the country and what it is of a tourism nature that could suddenly spur your interest just three weeks after you left the country.

"to see friends again" is also unconvincing. You can be expected to have had friends in your home country – that in September you had not seen for more than a year (unless they had visited you in USA or you had made one or more return visits during that time or met up elsewhere). Allowing a mere three weeks for them (after a year's absence) before returning to USA for two months for other friends (after three week's absence) seems unbalanced.

"to see my boyfriend" seems very plausible but may count against you. After such a short absence It suggests your ties to him may be stronger than your ties to your home country. This is ominous as far as visits are concerned because it seems he is resident in USA hence could be reason for you "going underground" there and becoming an illegal immigrant, possibly working in USA to sustain yourself (hence in competition for employment with citizens of USA) and perhaps not paying your taxes as part of trying to be 'invisible' to the Government.

Do I need to show proof of places I'm staying or anything?

It is a routine requirement to provide details of where you expect to spend your first night in USA but evidence of that is very rarely requested.

Am I allowed to say that I'll be staying with my boyfriend?

Not only allowed but strongly encouraged to do so. While with enough chutzpah it must be possible to lie one's way through Immigration, the risk of any untruth, however seemingly innocuous, is sky high. What may appear a little 'white' lie that is "near enough" the truth still damages credibility. Any inconsistency could be 'fatal' because it makes everything else you say of questionable veracity. Doubt is more than enough reason to deny you entry when, under the VWP, CBP don't need any reason to do so anyway.

HOWEVER, as @phoog has mentioned, It's allowed. You do stand a chance. IMO not enough to justify a transatlantic airfare to test for at least a little while longer, but an attempt at entry in not guaranteed to fail. I think waiting at least a few months would be required to reduce the risk significantly of being turned back.

You do seem to have other options and even if they do not suit you being prepared with reasons why not (at least for the first two) might help any case you have to put to CBP.

One is, at least this time around, have your boyfriend visit you rather than you visit him.

Another is to apply for a B-2 visa. This gives you a chance to make your case without flying across the Atlantic. If granted it is still no guarantee of admittance, that remains a decision for CBP, but with a B-2 you should not expect any problems at the border.

A third is to fly via Dublin where USA operates Preclearance procedures.