It is unlikely that you can prevent np endocytosis for such a long period of time. Typically, you can keep your cell suspension cold (4C) to prevent endocytosis, but this would be for a very short duration (minutes to hours, but you would want to monitor cell viability during the experiment).
Erika is right. It would be very unlikely that you can prevent the endocytosis of the nanoparticles into the cell, particularly for as long as 21 days.
There are a range of substances which can slow or inhibit endocytosis (brefeldin A, colchicine, filipin, sucrose) but this may cause you issues with cell viability and may result in poor adhesion between the particle and the cell membrane.
The other two researchers were quite right that free nanoparticles would most likely be endocytosed by cells unless they are thermodynamically trapped at 4C. The easiest way to label cell membrane is to use a lipophilic dye, such as Laurdan. If you have to label cell membranes with fluorescent nanoparticles, a rather unusual approach to my knowledge, then here is a suggestion: first graft your fluorescent nanoparticle to a lipid molecule such as zwitterionic DPPC or negatively charged DPPG to form a linear construct. If the nanoparticle is hydrophilic (such as polymeric nanoparticle) or negatively charged, it will stay free from endocytosis for a considerable amount of time. This is because the lipid moiety of your construct would partition into the membrane and energetically the nanoparticle would have no incentive to penetrate into the lipid bilayer. You could try this with model cell membranes/vesicles first and move onto real cells. The concentration of the nanoparticles depends on the sensitivity of your imaging device, and you could also find the ball park of that concentration by calculating the ratio of the cross section of your nanoparticles (pi* r^2*N) to the surface area of a cell (4 pi* r^2). If viability is a concern, make sure that ratio is less than 1/1000 so that the presence of the nanoparticles would not interfere with the lateral diffusion of the lipids in the cell membranes. The toxicity of the nanoparticles may be minimized by using polymeric materials or materials of net negatively charge. Positively charge nanoparticles should be avoided due to their high affinity for the net negatively charged cell membranes and their effect on the membrane electric potential.