I thought she should have thrown it further away, but it didn't surprise me at all that Claire didn't keep it. She's not familiar with guns, especially not a rifle like that guy was carrying, so she must have known that there was a considerable risk she would end up hurting herself of someone else with it by accident. Plus, as a nurse she might simply be against using guns on principle.
"We're a team, Garrus. There's no Shepard without Vakarian."