I resolve Reanimate targeting the Iona, Shield of Emeria in my graveyard. However, my opponent controls a Yixlid Jailer. Does Iona lose its replacement effect ("As Iona enters the battlefield…) and enter play without my naming a color?

You will choose a color for Iona, Shield of Emeria.

Replacement effects normally must exist before the event they attempt to replace occurs in order to work. However, replacement effects that modify how a permanent enters the battlefield need special treatment, because–among other reasons–tokens don't exist at all before entering the battlefield, but we still want token copies of Shambling Ghoul to enter the battlefield tapped, and similar.

So we have CR 614.12, which explains how to determine which replacement effects apply to an object entering the battlefield; there's a bunch of clauses in there for handling special cases, but I've bolded the part that's relevant to this question.

614.12. Some replacement effects modify how a permanent enters the battlefield. (See rules 614.1c–d.) Such effects may come from the permanent itself if they affect only that permanent (as opposed to a general subset of permanents that includes it). They may also come from other sources. To determine which replacement effects apply and how they apply, check the characteristics of the permanent as it would exist on the battlefield, taking into account replacement effects that have already modified how it enters the battlefield (see rule 616.1), continuous effects generated by the resolution of spells or abilities that changed the permanent’s characteristics on the stack (see rule 400.7a), and continuous effects from the permanent’s own static abilities, but ignoring continuous effects from any other source that would affect it.

So, to determine which replacement effects apply to Iona, you look at how it would exist on the battlefield, and on the battlefield, Yixlid Jailer doesn't matter anymore; Iona will have her “choose a color” ability as normal, so you will choose a color.