What regression model to use when independent variables are percentages to predict % outcome? "Independent" variables: time spent (% at work, % sleeping, % exercising), body mass composition (% fat, % muscle, % bone)
Dependent variable: Smoker (1) or Non-Smoker (0)
What kind of regression model should I use when subsets of the "independent" variables are percentages and are therefore not completely independent of each other?
 A: Your response is binary and so you probably want to look at something like a binomial GLM for that, such as logistic regression.
Having a group of $k$ predictors that add to 1 (e.g. the $k=3$ body proportion predictors) would imply that at most you can have $k-1$ of them in the model because of the multicollinearity issue.
However, I'm going to suggest that you may also want to transform those percentages; they're unlikely to enter the model linearly; indeed with a logit link my first thought would be that you might want to try something like the logit of the proportions instead.
A: I'd also go for logistic regression since you don't mention that you have a time variable specifying the time from inclusion in the study until start of smoking or censoring (end of study); in that case a Cox regression would be better.
I doubt that there is any difference in using percentages as predictors as compared to other continuous variables. For example, BMI (Body mass Index) is neither a directly measured predictors, as it is derived from a division of two units.
As Glen_b mentions, these predictors might not be truly linearly associated with the dependent variable. But transforming them might make predictors more difficult to interpret and journals typically don't like transformed variables if they are the predictors of main interest.
