No. Not all 5670's support Eyefinity. The one you linked is an example of a model that does not.
Eyefinity is only supported on cards that have a Display Port output. If it has VGA, such as the one you linked, Eyefinity is not supported. Display Port output will be accompanied by HDMI and DVI outputs as well.
Try this one instead: http://www.newegg.com/Product/Product.aspx?Item=N82E16814102870
If you do not have a Display Port monitor, you can get adapters that will allow you to utilize Eyefinity. For more information on them, see this article