Is there any evidence that Rambam was familiar with the writings of R' Yehuda Halevi? The more concrete the better.
The most concrete evidence that the Rambam knew of Yehuda Halevi is that Yonatan of Lunel, in a letter to him, listed the Kuzari as one of the books Yehuda Ibn Tibbon translated from Judeo-Arabic:
הרב ר' יהודה אבן תיבון הספרדי , אשר הועילנו והשכילנו ולימדנו מספרי החכמות , בהעתיקו אלינו ספר האמונות , וספר חובות הלבבות , ומידות הנפש ומבחר הפנינים , וספר הכוזרי
The Rabbi Yehuda Ibn Tibbon, the Sephardi, who improved us and educated us and taught us from the books of wisdom by copying for us...the Book of the Kuzari...
In another letter, from the Cairo Geniza (T-S 10 K 8.14), a friend writes to the Rambam about missing him, and quotes poetry from Yehuda Halevi to express it. That poem (אחר גלות סוד מה אטמין) was originally written for another Moshe—that is, Moshe Ibn Ezra—and the letter presupposes the Rambam understanding the reference:
And as God put off the appointed time [for writing back], there came to mind the strophic poem having melody, "Why continue concealing when my secret is out," etc., which contains some of his traits coinciding with my yearning and love for him.
--- See Joel Kraemer, "Six Unpublished Maimonides Letters from the Cairo Geniza" (Maimonidean Studies, 1991)
Of course, scholars have devoted much ink to show how the Rambam's writings critically responds to Halevi's, but much of this is theoretical (and the Rambam never explicitly cited him). But, considering Halevi's documented fame in Egypt during the Rambam's time, and the above references, it's certain that the Rambam was at least familiar with his predecessor's poetic and philosophic work. To what extent, and how that might have affected him, is—for now—speculative.