The eggs are not strictly needed for the home made pasta.
It has to be underlined however that the taste, the colour and texture of the food will be different.
About the taste
Nothing to say or to do about it, with eggs and without eggs simply are two different things.
About the texture
I can suggest to try the durum wheat semolina (in original [1,2] the "farina di gran duro rimacinata").
It will give a different sensation when you will eat (different texture); moreover it will facilitate the task to obtain a better coloured pasta.
Around 70 g of semolina and 40 g of water (plus salt of course) per person, meanwhile the proportion for the egg version is of 100g of wheat flour for each egg...
About the colours
There are many choices that will enrich the original flavour, and that almost always have to be related with the sauce you intend to use with.
- red: tomato concentrate, 1 tbsp each 100g of flour. It's a classic, almost neutral. Good with everything.
- orange: 50g of carrots (boiled and pureed) of 100g of pumpkin for each 100g of flour. Note the result should be sweeten.
- yellow: saffron (around 0.6g) for 100g. One coffee spoon (a little less then a teaspoon) of Curcuma can be used instead. It will mark the taste.
- green: 10g of boiled spinach, (or of basil leaves) for 100g of flour. But even lettuce  or broccoli.
Another classic, almost neutral.
- blue It's possible to try to boil the Red cabbage for some minutes and to add one teaspoon of sodium bicarbonate to make it blue: it's used for cakes , you can try with pasta too.
Two spoons of liqueur similar to the blue Curaçao is a quick solution, but the blue comes from a food colouring (usually E133).
Nice the match with white sauces as the ones based on cooking cream, bechamel or mild cheeses.
- violet to pink: beet, 30g for 100g of flour for the violet, less for a colour more close to pink. Again nice the coupling with white sauces.
- black: squid ink. It gives a dark black colour. I report since it is another classic and you said to have no problem to eat meat and animal products from time to time... Strictly for fish sauce of course.
- brown: 1 teaspoon of cocoa (with no sugar!) each 100g of flour. It is suggested with (hunting) game, for the colour and the stronger taste that can cover the cocoa aftertaste.
Dried Porcini Mushrooms 25g for 100g of flour,rehydrate in hot water , strain and blend.
It will mark the final taste.
Often in cooking when you mix more colours together it comes out brown: give it a try.