Do personal checks expire in the US? Also, if I stamp the following on it:
VOID if not claimed within 60 days.
Will that be honored by most banks?
It depends on the bank. According to the Uniform Commercial Code, a bank is not obliged to pay a cheque after six months, but may do so if it wants to.
§ 4-404. BANK NOT OBLIGED TO PAY CHECK MORE THAN SIX MONTHS OLD.
A bank is under no obligation to a customer having a checking account to pay a check, other than a certified check, which is presented more than six months after its date, but it may charge its customer's account for a payment made thereafter in good faith.
As for your second question, if you stamp "void after 60 days" on your cheque; I don't have a specific answer for that part (yet).
Update: I can find no specific rules about someone putting an arbitrary "void after xxx days" on their personal check. Businesess are alllowed to, but again the overriding rule seems to be that after six months it's the bank's choice, and you certainly couldn't make a cheque expire before six months, so I don't think that putting a stamp would make any difference. It's still up to the bank in the end.
When I last asked a certain large bank in the US (in 2011 or 2012), they didn't offer expiring personal checks. (I think they did offer something like that for business customers.)
They also told me that, even if the payee cashes the check a year later and the check bounces, even if it's because I have closed the respective account, he will be able to go to the police and file a report against me for non-payment. (This is what the customer service rep told me on the phone after a bit of prodding, but someone else feel free to improve this answer and fix details or disagree; it's hard to believe and quite outrageous if true.)