Can a planet support life in a binary star system where there is no definite day-night cycles - but the nights are like bright twilight mornings?
The short answer is no, such a planet is not possible.
There are two possible configurations for a planet that's part of a binary star system. One is that the two stars are relatively close together, and the planet orbits them both - this is known as a "circumbinary" system. The other is that the stars are far enough apart that, for all intents and purposes, the planet orbits only one of them, and the other is a distant object. Any other configuration is unstable over long periods of time.
This answer explains how the mechanics of circumbinary systems work. By necessity, the two stars are close to one another in the planet's sky - this is the famous "binary sunset" image that you see in e.g. Tatooine in Star Wars. Your day might be a little longer, but you'd still have nights where neither sun is in the sky.
This answer goes into more detail about what you'd see in the other configuration. In order for the orbits to be stable, the second star has to be so far away that it's visible merely as a point of light, like another planet would be. Obviously that doesn't provide enough light to make your nights into days.
From a physics standpoint you have one real option here but it is rather unlikely. To make it so none of the planet is ever in a night time state the planet must always be between the two suns. The way to do this, but still unlikely option is that the planet sits in the L1 lagrange point. It could have a spin, or it could be tidally locked to one of the two suns. Either way it will essentially have day all the time. With the two sides facing the suns being at "noon" and "noon 2" and the two sides perpendicular to the suns being at a dusk state, but still having some sun. Based on the relative size of the two suns you could have "noon" be bright and "noon 2" be dimmer, or both be similar in strength. The overall size and brightness of the suns would determine this. Remember as you build the world out the planet will be closer to the smaller of the two suns if you go with this option. This is unlikely as things do tend to drift and become unstable over time in the L1 points, but in theory could maybe work. That said, a lot of people think of Lagrange points as stable, so you could write it into your story and not too many people are going to question it, it just depends on how strict to science you want to stay.
A more stable solution is to have the planet in the L4 or L5 location. The problem here being it doesn't completely fulfill your "never night" requirement. Based off relative size/brightness of the suns, and if you are okay with a tidally locked planet (I'm not sure this could happen, but might be possible) you could end up with a planet that had a good portion of it's surface always be in daylight or twilight, but that setting would never change. If the planet instead had a rotation you would end up with a day/night cycle, but the night time would be rather brief, with the details depending on the exact make up of the system.