NYIschremp44 had a list of five NHL teams in the last 20 years with no one having two-degree connection. 2008 Canadiens have since been eliminated because of Streit.
Actually, by looking the rosters, 2001 Sharks can also be eliminated, at least the end-of-season roster because of Teemu Selšnne. They were team-mates for about month after Selšnne was traded from Winnipeg to Anaheim and then Sillinger was traded from Anaheim to Vancouver.
In 2006 Sabres I don't see player who played only in that year, so all of that team should be within two.
I can say that I am 4 degrees away. I can classify myself within two degrees of Valtteri Filppula who is two degrees of Mike Sillinger himself, having not played with him yet.
I am 3 degrees away. I have played hockey with Stacy Roest. Roest played on the Red Wings in 98-99. Sillinger played on the Red Wings in 94-95. Players such as Yzerman, Fedorov and Lidstrom played with both Sillinger and Roest.
So it goes from me to Roest to Yzerman et al to Sillinger.