From the documentation:
Streams may or may not have a defined encounter order. Whether or not
  a stream has an encounter order depends on the source and the
  intermediate operations. Certain stream sources (such as List or
  arrays) are intrinsically ordered, whereas others (such as HashSet)
  are not. Some intermediate operations, such as sorted(), may impose an
  encounter order on an otherwise unordered stream, and others may
  render an ordered stream unordered, such as BaseStream.unordered().
  Further, some terminal operations may ignore encounter order, such as
  forEach().
If a stream is ordered, most operations are constrained to operate on
  the elements in their encounter order; if the source of a stream is a
  List containing [1, 2, 3], then the result of executing map(x -> x*2)
  must be [2, 4, 6]. However, if the source has no defined encounter
  order, then any permutation of the values [2, 4, 6] would be a valid
  result.
For sequential streams, the presence or absence of an encounter order
  does not affect performance, only determinism. If a stream is ordered,
  repeated execution of identical stream pipelines on an identical
  source will produce an identical result; if it is not ordered,
  repeated execution might produce different results.
For parallel streams, relaxing the ordering constraint can sometimes
  enable more efficient execution...
Also, as you mentioned that processing order matters for you see here:
Stream pipeline results may be nondeterministic or incorrect if the
  behavioral parameters to the stream operations are stateful... 
The best approach is to avoid stateful behavioral parameters to stream
  operations entirely; there is usually a way to restructure the stream
  pipeline to avoid statefulness.
See also the answer to this question: How to ensure order of processing in java8 streams?
In short, it looks like you can preserve the order, if you use sequential streams (streams that are executed in one thread) and if you are careful with operations like forEach(). However it probably is not a good idea.