Eglamour is a gentleman at the Duke's court in Milan. We don't hear much from Eglamour, but Silvia describes him as "Valiant, wise, remorseful, well accomplish'd" (4.3.3). Because he knows a thing or two about love and heartache, he agrees to help Silvia run away from Milan to find Valentine. When Silvia asks for his help, she reveals Eglamour's back-story:
Thyself hast loved, and I have heard thee say
No grief did ever come so near thy heart
As when thy lady and thy true love died,
Upon whose grave thou vow'dst pure chastity. (4.3.20-23)
Eglamour sounds like a pretty loyal guy, don't you think? Apparently, when his "true love died," he vowed to remain celibate. (Notice how the Italian word for love appears in Eglamour's name?) Now, he's being a great friend to Julia by agreeing to escort her to the forest. This makes Eglamour a perfect foil to Proteus, who hopscotches between one love interest to another, attempts to rape Silvia, and stabs his best friend in the back.
But wait! It's possible that Eglamour is a coward. Critic Anne Barton points out that "Sir Eglamour, the romantic knight faithful unto death, abruptly turns coward and abandons Silvia to her fate when they meet the outlaws in the forest (The Riverside Shakespeare, 179). Hmm. Come to think of it, Barton may be right. After the outlaws capture Julia, the Third Outlaw notes that Eglamour, "being nimble-footed […] hath outrun" them (5.3.7). In other words, when the outlaws chased Eglamour and Silvia, Eglamour hightailed it out of there and left Silvia behind.