The first reason is that Andruw Jones beat up on left-handed pitchers to a tune of .286/.384/.540 last season, and for his career he's done much of the same (.263/.363/.504). Andruw, in my mind, is a no-brainer to start at DH against every left-handed starters.
What does that have to do with Johnny and Hideki? Well, they would start as the DH against right-handed pitchers. Well, last season Damon hit .255/.314/.401 vs. RHP, which is not worth even $2 million in my opinion. Last season Matsui hit .242/.318/.336 vs. RHP, which isn't very impressive either.
Add the fact that neither Damon or Matsui can play the field, or should I say... play the field well at all, and I think they'd be a waste of a roster spot.
The other name that has been mentioned, and somebody the Yankees have shown interest in in the past is Carlos Pena. Well I'd be good with him, because even though Carlos' days in the field are really behind him too, Pena did hit .255/.388/.504 vs. RHP. The only problem with Carlos Pena is price, as I don't think he'd take a 1 year contract for around $2 million, seeing as how he made $10 million with the Cubs last season, and put up an OPS+ of 123.
I looked at the rest of the free agents and they either didn't hit that well against RHPs, or they didn't hit well enough to make up for the fact they couldn't play the infield, particularly 3B, which is what the Yankees should be thinking about right now (they already have a 4th OFer in Jones, a super-utility in Nunez, and a backup catcher in Cervelli). Which leads me to think the Yankees should either allow Brandon Laird to take that last bench spot, or bring back Eric Chavez.
Chavez hit .255/.322/.365 vs. RHPs last season, which is nothing to brag about, but over his career he has hit right-handed pitchers to the tune of .278/.357/.509, so there's precedent to believe he could get the job done.
So sign Pena if he'll come cheap, but if that won't happen (which I don't think it will) then they should go with Eric Chavez or Brandon Laird.