If you rechamber to .300 Win Mag, you will need to open the bolt face, as was already mentioned. I thought about doing a similar thing with an '06 - convert to .375 H&H. My gunsmith that opening up the bolt face wasn't a problem, but he recommended installing a Sako-style extractor.

As far as the magazine box goes, I don't think you will need to do anything to it. A full length .375 H&H round fit into the magazine on my Remington 700 (.30-06) as it was. Nothing needed to be changed there.