that's not really overwintering, it is just growing in the winter time. i do it all the time, though not with annuum type. actually, that is a wee lie, i have 1 superchili plant and a pequin that were in soil, then i moved them into my homemade dwc. the pequin has not done much but the superchili is gigantic, loaded with flowers and has to be moved back into soil as it is overtaking my dwc and overshadowing my other plants. my problem is i have now where for it once it hits soil.
the only advice, the aerogarden 7 is the 7 slot growing station and depending on which type of ornamentals you get, 7 plants would quickly overtake the unit and the roots will quickly tangle in the base. my homemade units are made out of plastic totes and approximatel twice to three times the size of the a/g 7. have have room for 8 plants per bucket but can only house 6 and even that is just too many plants for the size of totes.
i just cleaned out a neighbours aerogarden, they grew herbs and left for a 10 day holiday and asked that i look after their cats and plants. i decided to harvest their herbs as they had shut the machine off and neglected the plants. i lifted the lid to look at the roots and what a mess, there was literally no room for water, i don't see how they didn't burn the unit out.
i often check the plant roots and sometimes crop them back to encourage new side root growth.
good growing.