Hi, I just joined your virtual pet site and am finding it quite fun!
However, as I explore through the worlds I've had several notices that I
 have found an egg, and must 'click here' to pick it up - but, no matter
 how many times I click on the link, I cannot pick it up!! What am I
doing wrong?

You have to click that tiny little image/arrow type thing next to the "Click Here To Pick It Up!" Thing. I've also have issues since I run on a slow connection, and for some reason it doesn't always like that.

The new system frustrates me too. xD Hopefully, they'll change it back to something easier like we had in V2.

Thanks! Just tried that, and it finally worked! Yay!