Removing them could be a terrible job, unless your magic tool is good at scraping up old, dried glue (much heavier than regular 'glue'), and once that was tried, it's likely you'd need a rented sander to finish the job. You haven't said how many tiles are missing or in what location (a couple of edge pieces vs various centrally located ones). Normally you wouldn't have to remove a vinyl floor them to put something else on top unless the tiles had curled, so the sales people weren't giving you a 'story' about that. Have you the headroom to put down a new subfloor of 3/4" plywood?

You might consider ditra as an underlayment over the existing tile as long as they are well adhered.

Here is a link that might be useful: ditra

I don't know I would take advice from two strangers I met at a big box store (especially a sales associate not representing his employer). Carpet still may be a reasonable option.

For carpet, carpet tiles or another vinyl floor you don't need to worry about the old floor.

For ceramic tile the vinyl will need to be removed or a suitable substrate put over top it and fastened to the floor. Removing the old tile maybe very easy or very hard depending on the specific circumstances. I've had them pop right off or be a complete bear. You can always try removing a few and see how hard it is.

I would just get qoutes from a few installers and compare total costs.

I actually have carpet, ceramic and vinyl tiles in my basement.

You need to get whatever floor YOU want. Personally, tile in basement will be really easy to take care of. But it is colder. Carpet would be nice and warm but get dirty quicker. Carpet would be easier to replace if needed, tile not so easy. Lots of things to consider.

Salesperson comments seem odd. I would get several opinions. I do think the tile removal will save you some money but it can be hard and timeconsuming. For me, we do everything no matter how hard, we just allot for the extra time so as to spare our bodies.

Unless you intend to use the tiled squares, I would not use carpet in the basement. The foam cushion underneath absorbs humidity and will create a musky smell. If tiled carpet doesn't appeal to you, I'd recommend you lay some vinyl tile which is inexpensive and then put a large rug over it.

There is a chance that what the sales associate is telling you may be true, but just to be sure, I'd call someone on a professional level and just ask some general questions, they should be able to give you a good idea of what you can expect over the phone.

that's alot of work ,removing all those tiles 441 sqft

I would research whether moisture/water would not be an issue when installing tile in a basement. I would talk to a professional about that, as you may need a moisture barrier and specific products. Otherwise you may have tiles popping off with poor adhesion due to moisture coming up through the concrete slab.

With a large area, you also need to think about expansion joints beyond those around the perimeter. The maximum span of tile, before needing an expansion joint, varies depending on site-specific moisture and thermal conditions.

You can ask over on John Bridge Tile Forum about your project. There are professionals who will be able to direct and advise you in figuring out your project.