I think Brian made a strong enough argument (albeit without references) that these superstitions are likely bogus, so I am going to focus on their origins.
Edit (over a year after I first wrote this answer): Brian's answer that I referenced above has since been deleted. In retrospect (thanks to Sklivvz digging it up from the archives), that other answer turned out to not have any properly referenced statements and was generally hand-wavy. The gist of the argument was that since many women go out in public these days while pregnant, there is likely a significant number of women who are inadvertently exposed to eclipses, yet despite this we don't see a very high rate of birth defects. The other answer was here before mine, so I probably just referenced it out of courtesy ;-)
It seems like at least some of these beliefs are rooted in traditional Mexican medicine. That article is in Spanish (and I don't have access to the full copy), but here is a translation of a portion of the abstract:
An analysis of the logic of one of the commonest health beliefs in rural areas of Mexico is made, taking as a starting point testimonies collected in the area of Ocuituco, in the state of Morelos. This belief suggests that a pregnant woman is in danger of having a harelipped baby during a solar eclipse. The interrelation of these beliefs with other traditional elements (such as the "loss of the shadow" and the "hot-cold theory") is discussed. Also, some of the already existing interpretations of this belief which seek to link the "loss of the shadow" with the solar eclipse belief are reviewed. Finally, an alternative interpretation of this belief is made from a structuralist methodological perspective. This interpretation is grounded in the Nahuatl myth on the creation of the sun and the moon, and in an analysis of the nature of rabbits in the Nahuatl culture, according to historic secondary sources. It is suggested that the belief about the danger of a solar eclipse must be interpreted in connection to the "hot-cold theory", but not to the "loss of the shadow".
"Loss of the shadow", or "la pérdida de la sombra" is a traditional belief that one's "shadow" (symbolizing the soul) can be separated from one's body. Hispanic "hot-cold theory" is based on the belief that some diseases/ailments are "hot" and others are "cold" in nature. Therefore, to summarize, this article is claiming that the avoidance of the eclipse among pregnant women was perhaps precipitated by a belief that it could strip one of one's soul and/or that its "hot" or "coldness" could instigate a disease.
The avoidance of eclipses among pregnant women is also apparently a custom in India, too. This study by Jain examined a medical center in India which had a higher-than-average rate of children born with missing limbs. The study was unable to conclude any correlation between the environment and the high rate of birth defects, however, it did note that
An interesting finding was a history of exposure of expectant mother to the eclipse during pregnancy in 19 cases [out of 200]. Though no documentary evidence exists to correlate it with the deficiency in the literature, still many parents believe this to be the cause of limb deficiency in the new born.