Can I use a 10s chain on a 3/32 cog for my singlespeed (not fixed)?

I wouldn't.
Use a cheap 1/8" BMX or 8 speed chain instead......much tougher, IMO.

I broke a 9spd chain into 3 pieces going up a really steep hill on my fixie. I haven't done the same hill since putting on an 8 spd chain, but it is much sturdier. Spend the $12 on a new chain and make sure to keep a few extra links in your bag. Lesson learned for me. Single speeds & fixies require beefy chains.

Put on a KMC 1/8" chain. Cheap and strong.

I'm using a $15.00 8 speed sram chain.

In addition to the strength and cost issues noted, there's another functional concern. Though nominally 3/32", cogs vary in actual thickness, and some are wide enough to bind and not work smoothly in a narrow 10-speed chain.

You may be trying to save $ by re-using a chain you have around, but you're better off to get a cheap chain that will work better.

Here's the KMC mentioned above, on sale at Nashbar for five bucks -- even comes with a quick link.
http://www.nashbar.com/profile.cfm?category=600076&subcategory=60001099&brand=&sku=12652&storetype=&estoreid=&pagename=Shop%20by%20Subcat%3A%20Chains

Just so I'm clear on this before I go making a foolish mistake, other than possible noise, is there any problem with continuing to use my 1/8" chain if my chainring and freewheel are changed to parts that happen to be 3/32"?

Just so I'm clear on this before I go making a foolish mistake, other than possible noise, is there any problem with continuing to use my 1/8" chain if my chainring and freewheel are changed to parts that happen to be 3/32"?

no problem at all