Let's first look at the word shalit. It derives from the root שלט, meaning "to rule". Klein provides the following etymology:
borrowed from Biblical Aramaic שלט (= he ruled, was master of, had dominion), which is related to Jewish Palestinian Aramaic and Syriac שלט (of same meaning), Ugaritic shlt (=ruler), Arabic saluta (he overcame, prevailed), Akkadian shalatu (=to rule), shaltu, shitlutu (= powerful, mighty), Ethiopian shallata (=he gave power.)
From this root we get the words shilton שלטון - "authority, government", sholtan שָׁלְטָן - "dominion" (note the kamatz katan, as this word that appears in the Rosh Hashana and Yom Kippur prayers - שהשלטן לפניך - is often mispronounced), and shlita שליטה - "control" (the honorific for living rabbis שליט"א is not related, but is an acronym of שיחיה לאורך ימים טובים אמן - "may he live many long and good days, Amen".)
Another related word is the Arabic sultan. The Online Etymology Dictionary has this entry:
What about shelet - "sign"? The word appears in the Bible six times, always in the plural, and always relating to military matters. The exact meaning, however, is not clear. Some (Rashi on Shir Hashirim 4:4, based on the usage in Yirmiyahu 51:11) say it means the quiver that holds the arrows, and others say it means "shield", perhaps specifically of leather (Ibn Ezra; Targum Yonatan on Divrei Hayamim I 18:7 and Divrei Hayamim II 23:9). Ben Yehuda writes that as a military term, shelet is related to שלט in the sense of power and might.
1550s, from M.Fr. sultan "ruler of Turkey" (16c.), from Arabic sultan "ruler, king, queen, power, dominion," from Aramaic shultana "power," from shelet "have power." His wife, mother, daughter, concubine, or sister is a sultana.
The German word for shield is "schild", and this apparently led to the word shelet taking on the meaning sign. Klein writes:
Borrowed from שלט shelet = (shield), on analogy of the homophonic German Schild, which has both meanings 'shield', and 'signboard' (however, there the two meanings are artificially differentiated inasmuch as Schild in the sense 'shield' is masculine, in the sense 'signboard' it is neuter).
Philologos, in this interesting article discussing why the Magen David מגן דוד is translated as "Star of David" instead of "Shield of David" notes:
Why did medieval Jews change David’s star to a shield? The obvious answer is that whereas stars have no great resonance in Jewish religious tradition, shields do. In numerous passages in the Bible, God is referred to as the shield of those who trust in Him, including more than a dozen times in the book of Psalms, of which the supposed author was David himself. And in the Songs of Songs we have the verse, referring to the mail-like plates in the necklace of the poem’s beloved, “Thy neck is like the tower of David built for an armory, whereon there hang a thousand bucklers [magen], all shields of [shiltei, the plural possessive of shelet] mighty men.”
It is undoubtedly a pure coincidence that shelet, which is a biblical synonym for magen, sounds very much like the English “shield” and the German and Yiddish Schild, which means both “shield” and “coat of arms” or “sign.” (It was under the influence of Schild, in fact, that shelet came to mean a store or street sign in contemporary Hebrew.) But can it be that the double-triangled hexagon, which was adopted by Yiddish-speaking Prague Jews as the emblem of their flag, was first called by these Jews of Prague a Schild, in the sense of a coat of arms, and then translated into Hebrew as magen and re-interpreted as the warrior’s shield that protected David? I wouldn’t rule out this possibility
So yes, the words shalit and shelet are connected. However, I think it's a good thing that we no longer need to associate Gilad Shalit with the signs requesting his release, but rather with the shielding and protection he finally received. May he live many long and good days, Amen!