I just looked through my Collins files again. I had forgotten that when I was at the National Archives I had hastily copied some records for a Joseph Collins who was in Co. B, 15th (Northwest) Arkansas Infantry and who enlisted July 10, 1861 at Bentonville, Arkansas. That's also one of the people who came up on Ancestry.com. This is clearly not my Joseph A. Collins, as he mustered out before my ancestor's letter requesting furlough was written.
I'm still puzzled about Joseph's remark about having 30 days' furlough coming because he was an "old soldier." He wasn't old in age; if the written family history I have is correct, he was born "about 1838," so he'd only have been 24 in 1862; and if he just enlisted in June 1862, his letter was written only about 3.5 months after his enlistment, so he wouldn't be "old" in length of service.
Is it possible he'd already served before and that his enlistment in June 1862 was a reinlistment? Perhaps if I order his record, I'll find out.
Thanks for your help, Bryan!