About those Senate rules: Could Cruz, Lee amend any Democrat bill to change Obamacare?

There have been many reports in the MSM that several Democrat senators (primarily the ones from Red states up for re-election) will very soon introduce a bill(s) to propose changes/delays/whatever in Obamacare, in a desperate attempt to curry favor with an electorate upset that they voted for the monstrosity in the first place.

Senate Republicans have long complained how Reid has run the place, especially his refusal to allow many, if any GOP amendments to bills.

I don’t  understand the Senate rules, but surely, Reid will have to bring this bill to the floor, otherwise he’d face a mutiny in his own caucus. And doesn’t that mean there has to be at least one GOP amendment offered?

If so, why not add an amendment to repeal Obamacare?

Let’s put the Democrats in a hard place, and squeeze ’em.