Colored string in mode line

@NickD provided a good answer: use a face.

OP's comment to Nick's answer says that he'll try to write a function that, given a string, returns a propertized string. Such functions already exist: propertize does that, and so does add-face-text-property.

For example:

(setq ss  (propertize "abcde" 'face '(:foreground "red")))

or

(setq ss  "abcde")
(add-face-text-property 0 (length ss) '(:foreground "red") nil ss)

You need to create a string with the appropriate face. You do that by attaching a face text property to the string, giving it value of some face (predefined or defined for the specific purpose - you can look at all the predefined faces with M-x list-faces-display and pick one from there, or you can define your own face). E.g. here's a snippet to use a predefined face:

(setq s (copy-sequence "sring"))
(put-text-property 0 6 'face 'custom-invalid s)
(setq mode-line-end-spaces s)