Do your cats eat bugs?

My cat seeks out the bugs that stray into our home but does not eat them. He plays with them and kill them. He has been indoors his whole life so I fear that he might get sick from eating bugs too. I am always trying to find the bugs before he can lay his paws on them. I do not like dealing with bugs and try to shoo them out out of the house with whatever I have handy,usually newspapers.