I do it a lot like Journey, but a little different. I freeze tomatoes until I have enough for a batch then put fresh ones and the frozen ones in the stock pot and start cooking them down. If you thaw frozen tomatoes the skin will slip right off but I don't bother too much with that. Before I freeze them I usually quarter and core them and cut out any bad spots I don't want in my sauce, like split areas or places where something has been nibbling on the tomato and it has gone off in that immediate area. There is usually a lot of good tomato left after trimming.
While you are cooking them down, most of the skins will separate from the tomatoes and will work their way to the top while you are stirring the pot to keep them from sticking. If you are careful, you can remove most of the skins before you put them through a food mill but be careful. It does get hot. I cook them down until most ofthe skin has separated before I run them through the food mill. The food mill will get the ones you miss, but if you can remove most of them you spend less time cleaning out your food mill. After I've removed the skins and seeds with the mill, I put them back into the stock pot to cook down. When you get to a catsup consistency, that is puree. You can keep going to get to a sauce consistency if you want, but keep stirring to keep it from sticking. I'm not sure how the Squeezo works with hot food since it has rubber and nylon parts. I'd be a bit concerned with hot food in that thing, but I am not familiar with it.
Since you are using water bath instead of pressure canning, make sure you add some acid to get the acid levels high enough to be safe. Not all tomatoes have a high enough acid content to be safe using water bath. Those low acid tomatoes are great for people with certain stomach problems but they can pose a risk when water bath canning.
I use all my tomatoes, not just the sauce tomatoes. You do have to cook them down a bit more, but I think I get a depth of flavor in the sauce that you just don't get just using paste tomatoes.
