File/Print changes footer

Instead of trying to track down the cause of the disappearance, it will probably be quicker to remove the table -- which is not necessary -- and just use tabs to position the fields.

The Footer style that's applied by default to footers (and the equivalent Header style) includes a center-aligned tab stop halfway between the margins and a right-aligned tab stop at the right margin, so no setup should be needed.

With the cursor at the left margin, use Insert > Quick Parts > Field > FileName to insert the FILENAME field. Press press the Tab key twice to get to the right-aligned tab stop, and press Alt+Shift+P to insert the PAGE field for the page numbers. 

The result should be stable as you print. You can also select the footer paragraph and add it to the Footer gallery (Header & Footer Tools > Footer > Add selection to gallery) for later use.

<Edited to get the fields in the correct positions, according to the first post in the thread>

I don't know that you need to avoid tables in headers and footers completely. Lots of people use them with no problems, and there are certain arrangements that are difficult to achieve without tables.

I suspect that your particular document had some minor data corruption that was causing the disappearance, or that you have an add-in that's interfering. Trying to track those things to the source can be a very time-consuming process. Because the footer you want is so simple, it's just easier to go with tabs only.