Here's what jumps out at me (fairly screams there is something we aren't telling you that will point out that this is crappy research): In their chart of all 50 state's rate of gun deaths, they do not show the percentage of gun ownership. They only report the percentage of gun ownership in the state for the top and bottom 5 states. That eliminates the ability to perform more analysis or even do a quick statistical check to see if their conclusion is over generalized (which I can pretty much guarantee you).
Furthermore, since they are calling it "gun deaths" it is for sure that they are including all firearm related deaths. They are not making any distinctions between or controlling for suicide rates, gang activities, legal v illegal use of firearms, etc. It doesn't even rise to the level of half-a$$ analysis. There is no analysis but rather only a reporting of 20% of general data.
Their conclusions were safe and foregone. Not only would I expect that gun deaths would be higher in places with more gun ownership, I also would expect that swimming pool drownings are higher in places with more swimming pools, traffic accidents are higher in places with more cars, motorcycle accidents are higher in places with more motorcycles and incidents of half-a$$ research with bad statistical analysis are higher in places with more tree-hugging liberals.