No, because of distance. I'm sure there is a simple equation that shows this but I don't know it. Presumably most stars we see are larger than earth, yet we can only just see them and they are basically point sources.
There is no distance at which a larger spherical star won't shine on over 50% of a smaller sphere's surface area. That includes distant stars and Earth.
Here's a rough diagram I sketched up in Autodesk Inventor:
We have a "planet" with diameter 0.275 unit and a "star" with diameter 1.0 unit. If their centers are 1.0 unit apart, then you can clearly see the extra part of the planet's circumference that is covered by the star's light. If you increase their distance to 10 units, the two lines connecting the circles get closer and closer to becoming parallel, but they will never become parallel because the star is larger than the planet.
This being said, the earth isn't a perfect sphere. There will be a distance where the topology of the earth will block distant stars from shining on >50% of the planet... for some of the day.
Here's a rough diagram I sketched up in Autodesk Inventor:
http://i.imgur.com/u9Xjqsa.png
We have a "planet" with diameter 0.275 unit and a "star" with diameter 1.0 unit. If their centers are 1.0 unit apart, then you can clearly see the extra part of the planet's circumference that is covered by the star's light. If you increase their distance to 10 units, the two lines connecting the circles get closer and closer to becoming parallel, but they will never become parallel because the star is larger than the planet.