I also have a '95 ZX-6R. When I got it the guy had replaced both front calipers with ones off an older ZX-9R I think. They worked but I wanted to get it back to original. I bought a pair of calipers off eBay and replaced the seals. They were like new. I also got the old original calipers with the bike. It turns out that they were O.K. just need some TLC. Anyway, you can get the bleeder valve from ronayers.com although I doubt that is the problem. You can get whole caliper but they are very expensive. Calipers come up on eBay every so often but only the '95-'97 are a perfect match.
I have never had any trouble bleeding the front. You might try putting some silicone grease on the bleeder and putting it back in. I noticed that mine were gunked up so I took them out and cleaned them up in ospho. There is a tiny hole on the side that tends to get clogged. When you bleed the brake pump the lever until it is firm and hold it down. Then just barely crack the bleeder, just enough to let the lever go down
I have never had any trouble bleeding the front. You might try putting some silicone grease on the bleeder and putting it back in. I noticed that mine were gunked up so I took them out and cleaned them up in ospho. There is a tiny hole on the side that tends to get clogged. When you bleed the brake pump the lever until it is firm and hold it down. Then just barely crack the bleeder, just enough to let the lever go down