Several answers suggest there's a solution if the species is a chameleon. If they aren't chameleons, however, they might use technology. It might be that they tattoo themselves with a material that binds to the blood vessel walls and exhibits this chameleon like behavior when the blood vessels dilate, exposing a red-colored pigment when stretched.

This is technically more simple than that of a chameleon because it only needs to generate one response to one stimulus. It only needs to get one color right, rather than a rainbow.

Drink Kool Aid

If you have a lot of silver in your system and you can't get rid of it fast enough, you become blue. This is a condition called Argyria and usually happens to people who take medications containing silver.

Your aliens can do something like that, but using a red coloring agent instead of a blue one. People on Earth are known to have dyed their hair red with Kool Aid (which makes me wonder what that poison does to your inner organs). Perhaps by drinking enough Kool Aid your aliens can dye their internal fluids red from inside out, thus being able to pass as a human when blushing.

If they can willingly alter their appeareance to a huge degree, then blushing a little should be second nature to them.

An example, you wouldn't ask how the T-1000 form Terminator 2 can blush. He's already making a liquid metal look like human skin, he can clearly make it look like human skin that's a little redder.

So if your shapeshifters are already changing form a creature with different colored skin or without something that resembles humas skin, they already have solved most of this problem.

Even if your shapeshifters have more modest abilities, there are a nomber of things they are probably already taking into account that would be a lot similar to blushing.

For instance, are they simulating skin marks, pimplies, freckles, other imperfectons?

Are they capable changing into people with different body types? Our skin donest look the same in every art of our body, look at out palms and the back of our hands, for instance. There's also the effect of the sun on the more exposed skin. Whenever a shapeshifter changes, they'd need the reight colored skin in the right locations, that mastery of their skin tone could be extended into blushing.

How they actually change the color of their skin, there are many posibilities. Manipulating pigments on theit skin would be an example, as EveryBitHelps mentioned. Then again, I would say it has to be something in tune to thier general shapeshifting abilities. If you already have that figured out, then adding blushing on top doesn't seem that complicated. If not, then I don't see why just focus o the blushing science.

Unless you are going for how to prevent an unconcious raise of blood pressure single them out. Then the shapeshifter would have to learn to identify the change and compensate. Or they could just shapeshif a thicker or more opaque skin. That never let's the underlying blood show through.

This would help other issues as well. If the undelying blood is contributing to the color of the skin, then the skin has a redder base tone by itsef. That means that when blood rushes out of an area (think when to press some part of your skin hard), it turns whiter in humans but it'd be the opposite effect for your guys.

The opposite of blue is red.

Unless you are looking at a pure frequency, for any color, the perceived color is the result of a mix of colors contributing to the end color.

Blushing means more blood in the skin and increase in contribution of red to the end color. The converse happens if someone is really cold with blood leaving the skin and going to the core. Without the red to balance, the (fair) skin of the cold person looks more blue.

For a blue blooded creature, removing a degree of blueness renders the skin relatively red. Consider this comely alien and her color inverse.

Assuming these shapeshifters are fair skinned at baseline (not black or brown or blue), to achieve this they must have a baseline balance of blue and red in their skin. Withdrawing some blue will result in a relative increase of red, just as in a cold fair skinned human withdrawal of some red results in a relative increase of blue.

It leaves open the question of why shapeshifters might not take on any color they choose. Presumably it must take some time to finish a shape, which is then difficult to modify on the fly by the same mechanism.

Chromotophores.

I'd imagine your shapeshifter has some chameleon like qualities in order to help them...um...shift!

Since chameleons can't generate their own body heat, changing the color of their skin is a way to maintain a favorable body temperature. ... Beneath this are several more layers of skin that contain specialized cells called chromatophores. The chromatophores at each level are filled with sacs of different kinds of pigment.

Source: Wired

Blushing would be second nature to them, they wouldn't need any special training in order to blush...maybe training in order not to blush too frequently.