Subset rows in data.table if all specified columns match a criterion

You can apply row-wise and then check if all the values in that row is greater than 0.

a[apply(a[, cols, with = FALSE], 1, function(x) all(x>0))]

#       n x y z
#1: case2 2 1 1

We can use Reduce with .SDcols. Specify the columns of interest in .SDcols, loop through the Subset of Data.table (.SD) check whether it is equal 0, get the sum of each row with Reduce, negate (!) to get a logical vector which returns TRUE when there are no 0 elements and use that to subset the rows of 'a'

a[a[, !Reduce(`+`, lapply(.SD, `<=`, 0)),.SDcols = cols]]
#       n x y z
#1: case2 2 1 1

Or as @Frank mentioned in the comments, pmin can be used as well

a[a[, do.call(pmin, .SD), .SDcols = cols]>0]

You can try

a[rowSums(a[, lapply(.SD, `<=`, 0), .SDcols=cols])==0]
#       n x y z
#1: case2 2 1 1

It selects the rows for which there are no cols columns with a value below or equal to zero (you can also use condition x>0 and check for ==length(cols) if you prefer).