The issue of negative values for "AgeDay1" is described in the "Nest survival models" chapter of the "Gentle Introduction to MARK" (currently on page 17-14) and that information should be relevant here. The text on the topic reads as follows.
Some of you may be concerned about the fact that negative values were entered for most nest ages on day 1 of the nesting season. But, the negative values are never actually used to compute a survival, because a nest doesn’t enter the likelihood until the ‘add(AgeDay1,x)’ value is zero, assuming that the correct age on day 1 (i.e., the correct negative value) has been entered. This business of adding to a negative age may be
conceptually difficult at first but it does work!
The key idea being that information for a nest is only used in the likelihood for intervals when it is under observation and the information for survival for that interval is provided.