No, I absolutely would not use sick time before quitting.
My employer is rather unique, in that it is still extremely generous (too generous) in sick time, vacation time and personal time. All employees receive:
17 paid holiday (I never knew there were 17 holidays until I started here)
Closed the weeks of Christmas and New Years (10 days)
160 hours of vacation time (80 for hourly employees)
160 hours of sick time (this can accrue up to 800 hours, and also functions as our company's short term disability policy; 66% LTD kicks in at 12 weeks, but you can keep your full salary for longer if you accrue more sick time)
16 hours of "personal time", and I've never understood the difference between this and vacation time
Half days on Fridays during the summer.
On the flip side, we are not paid out for anything if we leave the company (we have shockingly low turnover, however, and most people don't leave "willingly"). That includes not being paid out for vacation, sick or personal time. Because of this, none of the vacation time is kept on the books as a liability.
Also on the flip side, it is effectively impossible to use anything remotely close to your allotted vacation time and succeed in your job. I'll end this year (our corporate fiscal year ends Tuesday) with 80+ hours of unused vacation time and 550+ hour of sick time in the bank. I have never taken a sick day here, although I have take a few days off for sick kids and used vacation time for that.
Lastly, on the flip side, my team works insanely hard in a very high pressure industry, so they need some vacation time to stay sane. Most, like me, don't come close to using it all, but I encourage them to not let their work consume their lives, as a burnt out employee will cost me, and my employer, far more than somebody taking a week or two off to unwind.
ETA (because I'm waiting for a delayed flight home): The 10 days at Christmas include 2 of the paid "holidays", so the 17 paid holidays include three days at Thanksgiving, Christmas, New Years, President's Day, 2 days at July 4th, Memorial Day, Labor Day, Veterans Day, MLK, Valentine's Day (yes, that is correct, we are to "share the love" that day), Good Friday, Easter Monday, Columbus Day, and one that I'm apparently forgetting.