What color towel are you using? I think the colors suggested are very pretty. If you want to change the colors look at the stitch out on your software or computer and look for these things:
1. If you change a color what else is affected. For example, if you change the swan does it also change the flower?
2. Does a jump thread get stuck under another color. I've seen jumps that, if changed from a light to a dark color and if not snipped( or sometimes can't be snipped) show under a light color thread that stitched over it.

I had embroidered the letter C of the swanfont, on a bath towel, for the wedding of my GS.But the towel was dark purple, and the colour white of the swans and of the flowers stitched out very well.the swans and the flowers are white, and i found this well for the wedding. Here is the picture.
If you want absolutly the towel be white, there is black swans, and you could choose any colour for the flowers...

A selection of different shades of tans will look pretty and elegant on a white towel..... I know that lilies are usually white and so are swans.... but this is artistic license..... It would be very pretty.

MJDG

I did on white towels with gray colors for the embossing, swans, etc. If you have metroemb threads I used 1964 1969 1922 1907 non-metro 1907 non-metro black 2484 for the color changes. They turned out really beautiful.

i stitched for my daughter inlaw on a black pillow case last easter and it was beautiful but you may not see all the beautiful work on white

I have done the S on blue towels but think the white on white would be elegant. Why not try it on a small hand towel, if it looks good you'll have a bonus gift for them, if not you won't have ruined your big one.