In what circumstances does a mammalian egg copy its DNA?
The writers of the show may have been somewhat imprecise either by accident or intentionally to avoid excessive details.

There are several different stages of egg cells, with distinct names for each and for the process that leads from one to the next. The whole process of egg cell creation is called [Oogenesis].1 To quote Wikipedia:


  Oogenesis starts with the process of developing oogonia, which occurs
  via the transformation of primordial follicles into primary oocytes, a
  process called oocytogenesis.[4] Oocytogenesis is complete either
  before or shortly after birth.


To our current knowledge all primary Oocytes are present at birth. The next stage in the maturing process is called ootidogenesis, which produces an ootid. 


  The succeeding phase of ootidogenesis occurs when the primary oocyte
  develops into an ootid. This is achieved by the process of meiosis. In
  fact, a primary oocyte is, by its biological definition, a cell whose
  primary function is to divide by the process of meiosis.[8]
  
  However, although this process begins at prenatal age, it stops at
  prophase I. In late fetal life, all oocytes, still primary oocytes,
  have halted at this stage of development, called the dictyate. After
  menarche, these cells then continue to develop, although only a few do
  so every menstrual cycle.
  
  Meiosis I of ootidogenesis begins during embryonic development, but
  halts in the diplotene stage of prophase I until puberty. The mouse
  oocyte in the dictyate (prolonged diplotene) stage actively repairs
  DNA damage, whereas DNA repair is not detectable in the pre-dictyate
  (leptotene, zygotene and pachytene) stages of meiosis


DNA replication is completed before Meiosis occurs. So oocytes do not replicate DNA after birth, but they do repair DNA to fix mutations that may occour after birth but before conception. If DNA repair fails, then mutations may persist. So the show would appear to be wrong about egg cells copying DNA in an adult bear, but right in that mutations might occur during its lifetime affecting its offspring.