I tried to give Gannon the Q-35 Matter Modulator and give Lily the Oh,Baby! Super Sledge but they both refused to use them.I gave them both to Boone and he had no problem equipping them.

Only time I've ever had a companion use anything but what they are equipped with by default, is Cass. I gave her dynamite after she told me through dialogue that she likes dynamite . ...she used it. She won't use anything else I give her though. Aside from that.... Fallout 3 and New Vegas both I've given companions all kinds of weapons and they never bloody well use them. Clover used to drive me crazy. She kept saying "Can I have a bigger gun?" ...and I'm sayin ya, you moron, you've already got 4 of them!!! lol

I had to put my extra dynamite in ED-E after Boone started using it.