First of all, I like your idea. Have a look at other sites to see what they have done, some use Java others Windows media or Real players the choice is yours, click here animals page has a good selection of this type of site. I chose to use Windows media on my site but as I say the choice is yours but here is how I did it.
Go to microsoft.com and download free Windows Media Encoder 9 Series. This is the software that will allow you to stream from your webcam to your web page at a suitable bit rate, it is fairly self explanatory and easy to use. I stream at 25 kb/s which is not too demanding, suitable for my requirements (my webcam is submerged in my garden pond) and probably suitable for the bird box but you can always increase the bit rate. Note that this software automatically detects your IP address for you to cut and paste onto your embedded Windows mediaplayer on your web page. Next go back to microsoft.com and type in the search box - html code embed media player - this should give you a page with the correct coding to embed mediaplayer onto your webpage. You can alter the player to show or not show controls, autostart, window size etc.
There are limitations of course, a limited number of viewers can connect to your web site at any one time but on the other hand they can stay connected for as long as you are streaming. Viewers connect to the stream directly from your computer rather than the server but I have had no complaints from my ISP about using this method to stream my webcam.
I hope this has been of some help, no doubt there will be the usual mob shouting rubbish and giving a hundred reasons for doing it another way. Suffice to say that my underwater pondcam has been streaming eight hours a day for a year with no problems. When you do get it up and running register your webcam on webcam sites as it does bring in the visitors.