Should my Euonymus alatus 'Rudy Haag' (Burning Bush) be pruned in spring or not at all? This is my first spring after planting it last spring.
A winged burning bush. Sounds interesting. However, it is just a woody plant. Spring is the best time to prune!
If you like how it is growing, let it grow. If you want the stems to be thicker, let it grow. If you don't like how it is growing, prune it: remove stems you don't want, decapitate stems you want to 'split' into branches.
First thing in spring (so to speak) up to the time the buds are just breaking is a good time. All energy is stored locally as starch grains in living cells. What you have removed doesn't 'steal energy' from what you have left. The buds closest to where you cut will become new shoots; maybe the next one or two down the stem will too, but they almost certainly will pop leaves. A distal bud point in the direction the new shoot will go. If you don't like it, just rub that bud off or prune a little lower. If there is only one bud, the stem won't branch into two. It will just make the one stem abruptly change direction.
After bud burst you should not prune. You should let the new leaves open and 'harden'. At this point the foliage is now feeding the tree so that it can grow. Some of those sugars is also being stored (converted to starch grains) - the plant is 'building strength'. Certainly by the summer solstice, the plant has gained enough strength that you can prune again to shape the plant to be as you want without sacrificing its vigor.
Now it should be easy to appreciate that if the plant is growing too vigorously, you prune closer to the time the foliage on your plant hardens. Likewise you could also prune a third time after the solstice (i.e., in 'official' summer) to reduce vigor, though the degree to which this third occasion affects the plant varies by species.
Then you let it be and enjoy the wonderful fall color. After the leaves have dropped,it is easy to see the branch structure. You can correct this to your liking immediately or you can wait until when the buds begin to swell the next spring to prune it.
The wings, which may be your reason for choosing this variety, appear with age of the internodes. There will always be some new stem sections with no wings, and below those, internodes with two opposed wings, and below those, ones with 4 wings, and etc. Pruning or not doesn't change this.
Yes, of course you can prune Rudy Hagg burning bush. The only reason not to is that you like its natural growth habit and just want to let it grow and just 'do its own thing' (and there is nothing wrong with that).
There are very good reasons to NEVER PRUNE IN WINTER, but I suspect you are at least as tired of reading as I am writing and proofing this, so I'm not going to write anything further.
Enjoy your spindle tree. Don't be afraid to prune it! It will grow.