Eagle's Claw's effect is a +33% crit chance for, roughly, one attack that immediately follows it assuming it's queued to immediately fire. There is a precise time to it, but part of the buff is also wasted on the animation time so there's almost no chance of it impacting more than one ability. EC also has a base 15% crit chance against everything on its own.
Critical Strikes is +50% for roughly 3 seconds or so, I believe. Someone may be able to correct me on the exact time, but typically you can fit in 2 or 3 fast animating attacks.
So if you put the ATO into Eagle's Claw and had it proc your next immediate attack a minion would have a 88% crit chance. Against a lieutenant or higher, a 93% crit chance.
With the other passive Scrapper ATO in play, you'd be looking at 90%/97%. With the superior passive ATO, it's 91%/99%.
I think Eagle's Claw is a good spot for the proc in general; procs benefit from longer recharge times, cast times, and a low number of targets. Slapping it in there gives you a pretty high chance for it to activate, and I think the synergy for a near guaranteed follow-up crit makes the power worth taking despite the long animation.