It sounds like this person would be close to what you get with "pure lines", aka inbred strains in laboratory animals:
A population of identical homozygotes constitutes a pure line. Pure lines "breed true"; that is, they maintain their phenotype indefinitely if genetic crosses are restricted to members of the pure line (see True Breeding).
You get those by inbreeding for enough generations that the individuals become genetically identical, or monozygous for every allele at least.
And on the effect of being a homozygote for all genes:
Inbreeding in allogamic organisms bring the deleterious recessive alleles to homozygosity; the immediate consequence is an increase in the frequency of defective offspring, or, in another words, an increase in the genetic load of the population. This phenomenon is called inbreeding depression or inbreeding degeneration. As inbreeding continues, the deleterious alleles are selected out and eventually disappear. The original heterozygous populations are often more fit than the resulting pure lines because they profit from heterosis and balanced polymorphisms; the main advantage of pure lines is the quick production of many individuals with the same well-adapted genotype
It sounds like she would only have genetic problems if she had problem genes; bad alleles are more likely to be harmful if you're monozygotic for them because there are no other alleles to compensate, but if you only have the good alleles to begin with there's no harm. You would have problems in cases where having two different alleles is itself superior to having a single one of any; for things like immunity for example that can be the case. Or being heterozygous for the sickle-cell allele in malaria country, for another standard example.
I don't know if any of those things would be harmful enough that we'd call that woman "unhealthy" or "suffering from genetic problems" though. I think if it did, it would be more a statement about how the human genomes contains these specific alleles that you have to be heterozygous for than about the intrinsic harm of being the combination of two identical haploid genomes.
As for whether her egg cells would all be identical, they should be since homologous recombination should be switching identical bits of DNA with each other, but that's ignoring mutations. Which shouldn't be ignored, because every human has on the order of 100 mutations that weren't in their parents. Still, compared to how much DNA there is in the eggs overall it would also be reasonable to call them identical.