Q:

Do neutrons in a strong electric field show a small dipole behaviour?

As neutrons are neutral an electric field should not attract or repel them. However, as they are composed of a positive and a negative part, called one 'up' and two 'down' quarks, shouldn't those quarks somehow slightly react to that field making the neutron at least a mini electric dipole?

A:

Neutrons can indeed have a nonzero (permanent) electric dipole moment (same as electrons).

In the Standard Model of particle physics, this is predicted to have the value of $|d_n| \sim 10^{-31}\:e\rm \:cm$.
It is possible to test experimentally for this value to extremely high precision, but this is still short of detecting the SM value. Current experiments are limited to precisions of the order $\Delta d_n \sim 10^{-26}\:e\rm \:cm$, i.e., about five orders of magnitude larger than the expected value in the SM.

(The direction of the neutron's intrinsic EDM is strictly parallel to its spin.)
Improving these measurements down to a tighter precision is a high-priority area and the focus of active research efforts, because many extensions of the Standard Model predict higher EDM values than the SM for both the neutron and the electron; thus, getting a nonzero value of either quantity would give us a good idea of how the SM needs to be extended.

In addition to that, neutrons can indeed get a higher dipole moment if you place them in an electric field, i.e., they are polarizable. This is explained in more depth in a thread linked to in the comments. This is quite small, but it is nonzero, and it has been measured.

That said, you should be wary of this picture:

as they are composed of a positive and a negative part, called one 'up' and two 'down' quarks, shouldn't those quarks somehow slightly react to that field making the neutron at least a mini electric dipole?

The picture of one-up-and-two-down-quarks is a simplified picture of the neutron, and it has limited validity; moreover, to the extent that it does hold, it is subject to quantum mechanics, which tells us that the quarks never have well-defined positions inside the proton, much like the way electrons inside an atom occupy orbitals instead of well-defined positions. (Even worse, you can't even use the language of orbitals the way you can for electrons, because QCD is a highly-coupled, highly-correlated theory.)
Some of the classical intuition in terms of displacing charge distributions still holds for the neutron, but it needs to be taken with a grain of salt -- it is not quantitatively accurate, and the neutron is a highly nonclassical object if you look at it from up close.

