Question
It is a young birch and the "eggs" are all over the lower leaves.  what are they and will the tree live?

Answer
You may be describing aphid insects.  Look at them closely and see if they will move when disturbed slightly.  They will be green and round bodied and soft covering.  They will squish in other words.  They will make the leaves puckered and curl up.  Although they will not kill the tree, if you find this, you will need to treat with an insecticide.  I would treat the plant with Bayer Advanced Tree and Shrub Insecticide.  This product is mixed with water and poured on the soil around the base of the plant.  It is taken up systemically and will kill the bugs within a week or so.  It also provides protection from reinfestation for up to a year.  I hope this fixes the problem.  Good Luck.