There is a marvelous old book (19th Century if I recall correctly) where I learned Calculus the first time, called "Calculus Made Easy" by Sylvanus P. Thompson, and subtitled "What one fool can do another can". He explains that dx means a "little bit of x" and shows a square with sides x and x + dx and you can see why you can "ignore dx^2". Of course it isn't rigorous in any sense, but it uses differentials to get all the essential ideas of both differential and integral Calculus across quickly and smoothly. Needless to say, once I had absorbed all these essential ideas I went on to read more rigorous books where limits were introduced and used to make precise what I already understood well from this intuitive introduction. If I recall correctly Calculus Made Easy was republished some years back (Dover?) and was quite popular. I would suggest that you recommend it to your students, with appropriate caveats.
(Added later) I checked online and indeed there is a recent reprinting (available from Amazon and the other usual places). Moreover it has three new chapters written by the late great Martin Gardner aimed at the modern reader. I'm going to buy myself a copy!