Mindfulness is an important operational concept in the practice of meditation. Yet, it bears different meanings in different schools of meditation. I am not going to discuss that in this post. Here, I shall discuss the meaning of mindfulness as applied in doing tai chi square form and tai chi round form. As with meditation, different schools of tai chi tackle or treat mindfulness differently. No explanation, of course including mine, is definitive. With this preamble, the following is my view on the subject.
For beginning students, they should not overly concern with the concept (and execution) of mindfulness when they learn the tai chi form as a mind-body workout. The best way to learn the form for beginners, in my opinion, is to learn the form together with zhan zhuang and some tai chi chi-kung exercises (like Wu-style 24 styles tai chi nei gong, and/or special supplementary exercises targeted to a student's special needs). It is only when a student has achieved a certain level of progress in firstly Joint-Opening and secondly in Body-connectedness (with breathing controlling to facilitate connectedness) should he learn mindfulness the tai chi way.
I know some tai chi teachers have said that the square form is there in order to set a standard so that different tai chi practitioners will not deviate (too much) away from the standard (like "Received pronunciation RP" in spoken English). This perhaps might be one of the reasons but certainly not the major one, some tai chi teachers also have said that the tai-chi sign is a circle with two round curves (the two fish), and hence true tai chi form should be done in circular rounds, i.e. the round form. Again, this can be regarded as one of the reasons but certainly not the major one.
Square form is sometimes called Form-of-the-joints (関節拳）。The objective is to use (and condition) our joints for chi generation. Mindfulness here means a practitioners shall focus on (the areas around) his joints (primarily his major joints, shoulder and hip). In his mind he should be (or train himself to be) able to FEEL the opening and moving of his joints together with chi-generation arising therefrom. In order to do that effectively, he has to stop momentarily when his joints change to a new direction, as required by the form movements. While he has to find the point-of-maximum-resistance in his joints (and move to overcome such resistance), most of the time, his movement will be in a straight line. Hence, the external appearance of square form. Chi-wise, he will feel chi concentrated more around his major joints (some physical and mental endurance is required for progress - a sign of training effect); and therefore will not be conducive to moving meditation.
Round form presupposes some progress in square form for a practitioner. The reason is that without a certain level of effective joint opening, the round form cannot generate enough chi for mind-body workout purpose, and certainly cannot achieve the objective of further opening the joints and more powerful chi generation. While doing the round form, a practitioner shall focus on the efficient flow and balance of chi inside his body. With efficient flow and balance, his internal sensation will be "full of chi", and therefore conducive to moving meditation. In order to achieve effective whole body chi-balancing, heightened chi generation in the joints need to be dampened (i.e. not done in the square form way). And when chi is thus generated in the joints, such chi shall be mindfully made to flow efficiently to the rest of the body to achieve a state of body-connectedness (as an internal sensation). To achieve the above, the external form will be round.
Last remark: insightful readers might have noticed that, based on the above analysis, a mix of square and round movements (in different degrees and in different choice) can be used in a student's daily practice, based on his own unique mind-body condition - which is exactly what a seasoned practitioner should do!